| L(s) = 1 | − 2·3-s + 2·7-s − 9-s − 2·13-s + 2·17-s + 2·19-s − 4·21-s − 2·23-s + 6·27-s + 2·29-s − 4·31-s + 16·37-s + 4·39-s + 4·43-s − 16·47-s − 9·49-s − 4·51-s + 2·53-s − 4·57-s − 26·59-s + 4·61-s − 2·63-s − 2·67-s + 4·69-s + 8·71-s + 22·73-s − 4·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.755·7-s − 1/3·9-s − 0.554·13-s + 0.485·17-s + 0.458·19-s − 0.872·21-s − 0.417·23-s + 1.15·27-s + 0.371·29-s − 0.718·31-s + 2.63·37-s + 0.640·39-s + 0.609·43-s − 2.33·47-s − 9/7·49-s − 0.560·51-s + 0.274·53-s − 0.529·57-s − 3.38·59-s + 0.512·61-s − 0.251·63-s − 0.244·67-s + 0.481·69-s + 0.949·71-s + 2.57·73-s − 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 64 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 16 T + 130 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 99 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 26 T + 285 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 124 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22 T + 259 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 154 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 224 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67231863420945891258511979446, −7.61771084735817423262311289446, −6.85845465646839794967814182019, −6.51267623496176993291326644980, −6.22547758500510876648110098454, −6.11505944747201651727072693718, −5.46872027428920782492720767828, −5.26710488766648331407377673434, −4.99881897894251650911263695977, −4.70522009819694583292009766514, −4.16711607889211427801941163521, −3.90579103523269888896598860152, −3.20673599428109798944918002263, −2.94072556248092215428615500613, −2.49838564805811716779720436384, −1.98525114463207343413458412028, −1.25348210809507190295900154157, −1.13896275890434627643294934888, 0, 0,
1.13896275890434627643294934888, 1.25348210809507190295900154157, 1.98525114463207343413458412028, 2.49838564805811716779720436384, 2.94072556248092215428615500613, 3.20673599428109798944918002263, 3.90579103523269888896598860152, 4.16711607889211427801941163521, 4.70522009819694583292009766514, 4.99881897894251650911263695977, 5.26710488766648331407377673434, 5.46872027428920782492720767828, 6.11505944747201651727072693718, 6.22547758500510876648110098454, 6.51267623496176993291326644980, 6.85845465646839794967814182019, 7.61771084735817423262311289446, 7.67231863420945891258511979446
